In 1931, the mathematician and logician Kurt Gödel proved his incompleteness theorems, showing that any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. Further to that, for any consistent formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory. The essence of Penrose's argument is that while a formal proof system cannot, because of the theorem, prove its own incompleteness, Gödel-type results are provable by human mathematicians. He takes this disparity to mean that human mathematicians are not describable as formal proof systems and are not running an algorithm, so that the computational theory of mind is false, and computational approaches to artificial general intelligence are unfounded. (The argument was first given by Penrose in The Emperor's New Mind (1989) and is developed further in Shadows of The Mind. An earlier version of the argument was given by J. R. Lucas in 1959.[1] For this reason, the argument is sometimes called the Penrose-Lucas argument).

Penrose's idea is a type of objective collapse theory. In these theories the wavefunction is a physical wave, which undergoes wave function collapse as a physical process, with observers playing no special role. Penrose theorises that the wave function cannot be sustained in superposition beyond a certain energy difference between the quantum states. He gives an approximate value for this difference: a Planck mass worth of matter, which he calls the "'one-graviton' level".[2] He then hypothesizes that this energy difference causes the wave function to collapse to a single state, with a probability based on its amplitude in the original wave function, a procedure taken from standard quantum mechanics.

When he wrote his first consciousness book, The Emperor's New Mind in 1989, Penrose lacked a detailed proposal for how such quantum processes could be implemented in the brain. Subsequently, Hameroff read The Emperor's New Mind and suggested to Penrose that certain structures within brain cells (microtubules) were suitable candidate sites for quantum processing and ultimately for consciousness.[4][5] The Orch-OR theory arose from the co-operation of these two scientists and was developed in Penrose's second consciousness book Shadows of the Mind (1994).[6]

Hameroff's contribution to the theory derived from studying brain cells (neurons). His interest centred on the cytoskeleton, which provides an internal supportive structure for neurons, and particularly on the microtubules,[5] which are the important component of the cytoskeleton. As neuroscience has progressed, the role of the cytoskeleton and microtubules has assumed greater importance. In addition to providing a supportive structure for the cell, the known functions of the microtubules include transport of molecules, including neurotransmitter molecules bound for the synapses, and control of the cell's movement, growth and shape.[5]

Penrose's views on the human thought process are not widely accepted in scientific circles (Drew McDermott,[7]David Chalmers[8] and others). According to Marvin Minsky, because people can construe false ideas to be factual, the process of thinking is not limited to formal logic. Further, AI programs can also conclude that false statements are true, so error is not unique to humans. Another dissenter, Charles Seife, has said: "Penrose, the Oxford mathematician famous for his work on tiling the plane with various shapes, is one of a handful of scientists who believe that the ephemeral nature of consciousness suggests a quantum process."

In May 1995, Stanford mathematician Solomon Feferman attacked Penrose's approach on multiple grounds, including the mathematical validity of his Gödelian argument and theoretical background.[9] In 1996, Penrose offered a consolidated reply to many of the criticisms of "Shadows".[10]

John Searle criticises Penrose's appeal to Gödel as resting on the fallacy that all computational algorithms must be capable of mathematical description. As a counter-example, Searle cites the assignment of license plate numbers (LPN) to specific vehicle identification numbers (VIN), to register a vehicle. According to Searle, no mathematical function can be used to connect a known VIN with its LPN, but the process of assignment is quite simple—namely, "first come, first served"—and can be performed entirely by a computer.[11]

Penrose and Stuart Hameroff have constructed the Orch-OR theory in which human consciousness is the result of quantum gravity effects in microtubules. However, in 2000, Max Tegmark calculated in an article he published in Physical Review E[12] that the time scale of neuron firing and excitations in microtubules is slower than the decoherence time by a factor of at least 1010. Tegmark's article has been widely cited by critics of the Penrose-Hameroff hypothesis. The reception of the article is summed up by this statement in his support: "Physicists outside the fray, such as IBM's John Smolin, say the calculations confirm what they had suspected all along. 'We're not working with a brain that's near absolute zero. It's reasonably unlikely that the brain evolved quantum behavior', he says."[13] In other words, there is a missing link between physics and neuroscience,[14] and to date, it is too premature to claim that the Orch-OR hypothesis is right.

In response to Tegmark's claims, Hagan, Tuszynski and Hameroff[15][16] claimed that Tegmark did not address the Orch-OR model, but instead a model of his own construction. This involved superpositions of quanta separated by 24 nm rather than the much smaller separations stipulated for Orch-OR. As a result, Hameroff's group claimed a decoherence time seven orders of magnitude greater than Tegmark's, although still far below 25 ms. Hameroff's group also suggested that the Debye layer of counterions could screen thermal fluctuations, and that the surrounding actin gel might enhance the ordering of water, further screening noise. They also suggested that incoherent metabolic energy could further order water, and finally that the configuration of the microtubule lattice might be suitable for quantum error correction, a means of resisting quantum decoherence.

In 2007, Gregory S. Engel[who?] claimed that all arguments concerning the brain being "too warm and wet" have been dispelled, as multiple "warm and wet" quantum processes have been discovered.[17][18]

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